On stability analysis in shape optimisation : critical...

Control and Cybernetics

vol. 32 (2003) No. 3

On stability analysis in shape optimisation :critical shapes for Neumann problem1

by

M. Dambrine2, J. Soko lowski3 and A. Zochowski4

2Laboratoire de Mathematiques Appliquees de Compiegne,Universite de Technologie de Compiegne,

former affiliation : Antenne de Bretagne de l’EcoleNormale Superieure de Cachan where this work was done;

e-mail:[email protected] Elie Cartan, Laboratoire de Mathematiques,

Universite Henri Poincare Nancy I,B.P. 239, 54506 Vandoeuvre les Nancy Cedex,

France; e-mail: [email protected] Research Institute of the Polish Academy of Sciences,

ul. Newelska 6, 01-447 Warszawa, Poland;e-mail: [email protected]

Abstract: The stability issue of critical shapes for shape op-timization problems with the state function given by a solution tothe Neumann problem for the Laplace equation is considered. Tothis end, the properties of the shape Hessian evaluated at criticalshapes are analysed. First, it is proved that the stability cannotbe expected for the model problem. Then, the new estimates forthe shape Hessian are derived in order to overcome the classicaltwo norms-discrepancy well know in control problems, Malanowski(2001). In the context of shape optimization, the situation is similarcompared to control problems, actually, the shape Hessian can becoercive only in the norm strictly weaker with respect to the normof the second order differentiability of the shape functional. In addi-tion, it is shown that an appropriate regularization makes possiblethe stability of critical shapes.

Keywords: shape optimisation, stability analysis, shape gradi-ent, shape Hessian.

1Partially supported by the French–Polish research programme POLONIUM betweenINRIA–Lorraine and the Systems Research Institute of the Polish Academy of Sciences.

504 M. DAMBRINE, J. SOKO LOWSKI, A. ZOCHOWSKI

1. Introduction

The classical theory of calculus of variations connects the stability of a criticalpoint of the functional E with the study of the second order derivative of Eat this point. It turns out that in shape optimisation, one has to restrict theadmissible family of domains to regular shapes, C1 or C2 for example, in orderto easily define the shape derivatives. Since many shape functionals are knownto attain the extremal values for smooth shapes, the use of the differentialoptimisation theory is rather natural. The existence of an optimal shape can beobtained either by solving the associated Euler equation, or by the compactnessargument combined with an additional regularity of the optimal shapes. Anadditional question raised in connection with this approach is the following one,on which we focus our attention. Usually, if the second order shape derivativeis considered at a critical point and, furthermore, the shape Hessian is nonnegative, the Hessian is only coercive in a strictly weaker topology comparedwith the topology for which the differentiability can be proved. This difficulty isa general feature of optimisation problems in the infinite dimensional functionspaces setting, Malanowski (2001). In others words, the positivity of the Hessianturns out to be a necessary condition for any critical shape to be only a stablepoint i.e., to be a local strict minimum, but we can not know a priori if thiscondition is also sufficient or not. That definition of stability is natural and it ismuch stronger compared to any directional stability which would follow directlyfrom the posivity of the Hessian at the critical point. This difficulty is pointedout by J. Descloux (1990) for the specific example i.e., for the minimizationof the Dirichlet energy functional in two dimensions. In such a case the statefunction solves the Poisson equation with the Dirichlet boundary conditions.

This particular case has been studied by Dambrine and Pierre (2000), whohave shown that, in this case, the positivity of the second derivative is suffi-cient to insure the stability of the critical point. Then, in Dambrine (2000),the method has been extended to a wider class of problems by relaxing the re-quired assumptions on the spatial dimension, the differential operator and thefunctional. However, only the second order scalar elliptic equations with theDirichlet boundary conditions were considered. In the papers of Belov, Fuji(1997) and Eppier (2000) the second order optimality conditions are also con-sidered. In this work, we discuss the case of the Neumann boundary datum.This type of boundary conditions increases the difficulty of the study.

For the sake of simplicity, we consider the following model problem. Let fbe a given function in C∞0 (Rd) such that∫

Rd

f = 0. (1)

Let α and v0 denote two non-negative real numbers, and d ≥ 2 be the spacedimension. We define the class Od as the family of all open bounded subsets Ωof R with the boundaries ∂Ω such that

On stability analysis in shape optimisation 505

(i) ∂Ω is a C3,α manifold of dimension d− 1,

(ii) supp(f) ⊂ Ω and ∂Ω ∩ supp(f) = ∅,

(iii) V(Ω) = v0,where V is the restriction of the d-dimensional Lebesgue measure Ld to theclass Od. The C3,α topology on Od is considered in the paper. We explain lateron why this regularity is needed for the stability analysis. The deformationfields V, used in the framework of the speed method, Soko lowski and Zolesio(1992), are supposed to be sufficiently regular to preserve the admissible classOd, actually V = C3,α(Rd,Rd) is the sufficient regularity assumption for ourpurposes. Under the assumption (1) the Neumann problem −∆u = f in Ω,

∂nu = 0 on ∂Ω,(2)

admits the regular solution uΩ, which is defined up to an additive constant. Tomake the solution of (2) unique, we use the following normalisation condition∫

∂Ω

uΩ = 0 . (3)

We define the regularized energy shape functional Eσ on Od by

Eσ(Ω) =12

∫Ω

|∇uΩ|2 −∫

Ω

fuΩ + σHd−1(∂Ω),

= −12

∫Ω

|∇uΩ|2 + σHd−1(∂Ω).(4)

where Hd−1 denotes the (d − 1)−Hausdorff measure, so that Hd−1(∂Ω) is theperimeter of the smooth boundary ∂Ω, and σ ≥ 0 is the regularization param-eter. Since the gradient of uΩ is uniquely defined, the energy functional Eσ iswell-defined on Od even without any normalisation condition imposed on thestate function uΩ. We adress in this paper the following precise issue:

The identification of the conditions required for the specific shape op-timization problem which assure for the functional Eσ the existenceof a local strict maximum at the shape Ω∗ in the topology of Od.

Such a shape Ω∗ will be called a stable critical shape for the shape functionalEσ.

To simplify the presentation, we will focuss our attention on the energy shapefunctional E = E0, that is — the shape functional without the perimeter termwhich is in fact a regularisation term. The results of this paper remain valid incases of both σ = 0 and σ > 0. We mainly consider the question of stabilityof critical shapes and not the existence of optimal shapes. However, we do not


know if any critical shape Ω∗ exists for σ = 0. We can even present some nonexistence results which are given in Section 2.2. Let us point out that σ > 0 issufficient for the existence of an optimal shape.

Section 2 is devoted to the computation of the shape gradient DE of Eand to some remarks related to the topology of stable critical shapes. Actually,we prove that in some situations the critical shapes, if any, are unstable. InSection 3 the shape Hessian is evaluated at the critical shape and the question ofits definiteness is adressed. In particular, we prove that the situation describedabove actually occurs. The second order shape differentiability of the functionalE is obtained in the C2,α norm, however, the coercivity of the second derivativeD2E at any critical shape can only be expected, when it is the case, in the normof the fractional Sobolev space H1/2 on the boundary. Section 4 is the mainpart of this work, it includes the proof of the intermediate estimate∣∣e′′Θ(t)− e′′Θ(0)

∣∣∣ ≤ Cω(‖Θ− Id‖2,α)‖〈XΘ,n〉‖2H1/2(∂Ω0) . (5)

Here, ω is a modulus of continuity; Θ is an arbitrary C2,α perturbation of theidentity Id in Rd and eΘ denotes the restriction of the energy shape functional Eto the specific path connecting Ω0 with Θ(Ω0). Such a path is constructed usingthe flow of the given vector field XΘ. In other words, our construction is basedon the existence of the path, within the family of admissible shapes, connectingthe given shape Ω and its image obtained by the diffeomorphism Θ, the imagebeing denoted by Θ(Ω). The particular path can be parametrized by the familyof domains (Ωt)t∈[0,1] with Ω0 = Ω and Ω1 = Θ(Ω). For such a parametrization,the function eΘ(t) = E(Ωt) is defined over the interval [0, 1]. We keep in ournotation the dependence on Θ to emphasize the following crucial fact that themain estimate (5) is uniform with respect to Θ, for sufficiently small appropriatenorms of Θ. Section 5 is devoted to the study of Eσ, in particular, the existenceof a stable critical shape is proved. In Section 6, some of the formulae needed inSection 4 are established. Now, we can state the main result of this work. Wedenote by n the exterior unit normal vector field to the boundary of Ω0 ∈ Od.H stands for the tangent hyperplane, in the space of deformation vector fields,to the volume shape constraints at the shape Ω0.

Theorem 1.1 Let Ω0 ∈ Od be a critical shape of the energy functional E underthe volume constraint V(Ω) = v0, and let Λ∗ be the corresponding Lagrangemultiplier. If the shape Hessian D2LΛ∗(Ω0) of the Lagrangian LΛ∗ = E + Λ∗Vis negative definite, with the second order shape differential which satisfies thefollowing inequality

D2LΛ∗(Ω0)(V ,V ) ≤ −C‖〈V ,n〉‖2H1/2(∂Ω0) (6)

for some C > 0 and for all fields V ∈ H, then Ω0 is a stable critical shape forE in Od i.e., a local strict maximum of E at Ω0 occurs.


Remark 1.1 We make an intensive use of the shape calculus theory without re-calling all the details, we refer e.g., to the book by J. Soko lowski and J.P. Zolesio(1992) and to the technical report by F. Murat and J. Simon (1977) where thistheory is presented in the case of speed method and perturbation of identitymethod, respectively. We will use the speed method, but only with autonomousvector fields V , for the shape sensitivity analysis. Since we deal only with regularshapes, this assumption is not restrictive. We use the notations from Soko lowskiand Zolesio (1992). In particular, DE(Ω; V ) denotes the directional derivative(Eulerian semi-derivative) of the energy functional E evaluated at the domainΩ in the direction of the field V .

The validity of estimate (6) is discussed in Section 3. Note that Theorem1.1 is an easy consequence of estimate (5). Using the Taylor formula along thepath Ωt, we have

E(1) = E(0) +∫ 1

0

(1− t)e′′Θ(t)dt .

By the H1/2-coercivity assumption combined with (5) we have

e′′Θ(t) = e′′Θ(0)︸︷︷︸ + e′′Θ(t)− e′′Θ(0)︸︷︷︸,≤ −C‖〈V ,n〉‖2

H1/2(∂Ω0)≤ Cω(η)‖〈V ,n〉‖2

H1/2(∂Ω0)

Therefore, for η small enough, we get for all t ∈ [0, 1] :

e′′Θ(t) ≤ −C2‖〈V ,n〉‖2

H1/2(∂Ω0) < 0 ⇒ E(1) < E(0).

2. Shape derivatives and the resulting Euler equation

2.1. Shape gradients and the Euler-Lagrange equation

We use tangential differential operators, in particular, the tangential gradient∇τ and the tangential divergence divτ (·). For the sake of simplicity, we omit inour notation any explicit dependence of tangential operators on the boundaries∂Ω or ∂Ωt. Under the regularity assumption (i), it is known (see Soko lowskiand Zolesio, 1992, section 2.29) that E is differentiable in Od and that

∀V ∈ V,∀Ω ∈ Od, DE(Ω; V ) = −∫

Ω

〈∇u′,∇u〉 − 12

∫∂Ω

|∇u|2〈V ,n〉,

where u′ denotes the shape derivative of the state function u = uΩ, solutionof the boundary value problem (see Proposition 3-2 in Soko lowski and Zolesio,1992) −∆v = 0 in Ω,

∂nv = divτ (〈V ,n〉∇τu) on ∂Ω .(7)


We introduce the notation N := divτ (〈V ,n〉∇τu), used later on, especially inSection 4.

Remark 2.1 An integration by parts leads to∫∂Ω

divτ (〈V ,n〉∇τu) = −∫

∂Ω

〈V ,n〉〈∇τu,∇τ 1〉.

The integral in the right hand side vanishes since ∇τ 1 = 0. It means that theNeumann problem (7) fulfills the compability condition required for the boundarydatum, therefore, it is well posed.

Applying Green’s formula and the boundary conditions in (7), we obtain∫Ω

〈∇u′,∇u〉 =∫

∂Ω

u∂nu′ −

∫Ω

u∆u′ =∫

∂Ω

u divτ (〈V ,n〉∇τu) ,

= −∫

∂Ω

|∇τu|2〈V ,n〉 = −∫

∂Ω

|∇u|2〈V ,n〉.

Therefore, the first order shape derivative of the energy functional is given by

∀V ∈ V,∀Ω ∈ Od, DE(Ω; V ) =12

∫∂Ω

|∇u|2〈V ,n〉. (8)

Note that we have obtained for the shape gradient an expression depending onthe state function uΩ, but independent of the shape derivative u′ of the statefunction uΩ. This property is specific for the energy functionals and it is nolonger valid for any arbitrary elliptic shape functional, e.g., of the form

J(Ω) =∫

Ω

g(x, uΩ,∇uΩ) ,

where g is a given smooth function. Anyway, we can expect that the conclusionsof Theorem 1.1 can be extended for functionals of the above form, as it is thecase of shape functionals for the Laplace equation with the Dirichlet boundaryconditions (see Dambrine, Pierre, 2000, and Dambrine, 2000).

In the presence of the condition (iii) the shape optimisation problem underconsiderations is constrained. Hence, any critical point of shape functional solvesthe Euler equation for the Lagrangian LΛ defined on the family Od by

LΛ(Ω) = E(Ω) + ΛV(Ω) with Λ ∈ R \ 0. (9)

Therefore, the existence of a critical shape defined by solutions of the Eulerequation and denoted by Ω∗ requires the existence of the nontrivial multiplierΛ∗ 6= 0 such that the Euler equation

∀V ∈ V, DLΛ∗(Ω; V ) = 0 (10)


admits at least one solution in Od. The critical shape Ω∗ satisfies by definition

∀V ∈ V, DLΛ∗(Ω∗; V ) =∫

∂Ω∗

[12|∇uΩ∗ |2 + Λ∗

]〈V ,n〉 = 0 .

Since the first order shape derivative vanishes (10) for all vector fields in V, thecritical shape Ω∗ implies the additional boundary condition

12|∇uΩ∗ |2 + Λ∗ = 0 on ∂Ω∗. (11)

The function uΩ∗ solves the homogeneous Neumann problem, hence |∇uΩ∗ |2 =|∇τuΩ∗ |2 holds on ∂Ω∗ and

12|∇τuΩ∗ |2 + Λ∗ = 0 on ∂Ω∗. (12)

The Euler equation provides a simple characterisation of the boundary of anycritical shape Ω∗ as a geometrical domain, which has the property that theoverdetermined problem

−∆u = f in Ω∗,

∂nu = 0 on ∂Ω∗,

12 |∇τuΩ∗ |2 + Λ∗ = 0 on ∂Ω∗.

admits a solution.

2.2. Stability of domains determined by the Euler-Lagrange equa-tion

We can deduce from the first order necessary optimality conditions further in-formation on the topology of the optimal domain Ω∗ which is assumed to be astable critical shape.

Theorem 2.1 Assume that d = 2 or d = 3. If Ω∗ is a stable critical shape of Eσ

in the family Od, then any connected component of Ω∗ cannot be diffeomorphicto the unit sphere Sd−1.

Proof. We prove the theorem by contradiction, and distinguish the cases ofd = 2 and d = 3. First, we fix the notation. Assume that the domain Ω∗ hasa finite number of connected components (Ωi)i∈1,...,n. The boundary of Ωi isdenoted by ∂Ωi. We assume that the connected component denoted by Ω1 isdiffeomorphic to Sd−1 and, in this way, we obtain a contradiction.

First, let us consider the bidimensional case. The boundary ∂Ω1 of Ω1 is asingle Jordan curve of the length L > 0. Hence, there exists a function γ of theclass C3,α([0, 2],R2) such that


(γi) γ(0) = γ(L) ,

(γii) γ is one-to-one from [0, L) onto ∂Ω1,

(γiii) ∀s ∈ [0, L], ‖γ′(s)‖ = 1.

The tangential gradient of the solution is given by the derivative ddsuΩ∗ (γ (s)).

Equation (11) implies that∣∣∣∣ ddsuΩ∗ (γ(s))∣∣∣∣ =

√−2Λ∗.

From the elliptic regularity theory it follows that the trace of the solution uΩ∗

on ∂Ω1 is continuous. Hence, there exists ε ∈ −1, 1 such that

∀s ∈ [0, L],d

dsuΩ∗ (γ(s)) = ε

√−2Λ∗.

Since uΩ∗ (γ (0)) = uΩ∗ (γ (L)), it follows that for any connected component

0 = uΩ∗ (γi (0))−uΩ∗ (γi (|∂Ωi|)) =∫ |∂Ωi|

0

d

dsuΩ∗ (γi (s)) ds = εi

√−2Λ∗|∂Ωi|.

In other words, the Lagrange multiplier Λ∗ is trivial. Therefore, the volumeconstraint can be ignored and the Euler-Lagrange equation (10) reduces to theEuler equation

∀V ∈ V, DE(Ω∗; V ) = 0.

Moreover, we have on ∂Ω∗

∇uΩ∗ = 0 ⇒ uΩ∗ is constant on ∂Ω∗.

By the normalisation condition∫

Ω∗u = 0, this constant equals zero. Therefore,

Ω∗ is a C2,α domain such that for a function f there exists a solution of thefollowing problem

−∆u = f in Ω∗,

u = 0 on ∂Ω∗,

∂nu = 0 on ∂Ω∗.

(13)

We show that the solution uΩ∗ vanishes outside of the support of the right handside f . To this end, let us consider an open neighbourhood U of ∂Ω∗. By theHolmgren unique continuation theorem there exists an open neighbourhood Uof ∂Ω∗ such that the local problem

−∆u = 0 in U,

u = 0 on ∂Ω∗,

∂nu = 0 on ∂Ω∗.

(14)


admits a unique solution which is obviously the trivial solution u = 0. As Ucan always be restricted in such a way that U ∩ supp(f) = ∅, the solution uΩ∗

vanishes outside of the support of the right hand side f . Let Ω ∈ O2 be suchthat ∂Ω ⊂ U . Then, the open set U is as well a neighbourhood of ∂Ω and uΩ∗

solves also −∆u = f in Ω ,

∂nu = 0 on ∂Ω .

The Holmgren Theorem (see for example Courand and Hilbert, 1989, page 238)provides the local uniqueness result for the state function since the C2-boundary∂Ω∗ is not characteristic. In other words, uΩ∗ can be defined on the whole setU and uΩ∗ vanishes identically on U . Therefore, for all admissible domains Ωwith ∂Ω ⊂ U , and Ω \ U = Ω∗ \ U , we have

E(Ω) = −12

∫Ω

|∇uΩ∗ |2 = −12

∫Ω\U

|∇uΩ∗ |2 = −12

∫Ω∗|∇uΩ∗ |2 = E(Ω∗).

Hence E is locally constant around Ω∗. Therefore, the critical shape Ω∗ isunstable.

Let us now consider the tridimensionnal case. The same analysis remainsvalid except for the way to justify the triviality of the Lagrange multiplier andthe regularity of ∂Ω∗. The first order necessary optimality condition for ∂Ω∗

takes the form

12|∇τuΩ∗ |2 + Λ∗ = 0 on ∂Ω∗.

Therefore, ∇τuΩ∗ is a continuous vector field by the elliptic regularity theory,which is tangent by definition. If we assume that Ω∗ is the image of a ball by agiven diffeomorphism, then the tangent vector field has to vanish at some pointof ∂Ω∗ as stated by the Hairy Sphere Theorem. On the other hand, the normof the vector ∇τuΩ∗(x) equals −2Λ∗ for x ∈ ∂Ω∗, and the norm |∇τuΩ∗ | isconstant on ∂Ω∗. Therefore, the norm equals zero everywhere on ∂Ω∗. By theHolmgren theorem, uΩ∗ = 0 in an open neighbourhood of ∂Ω∗. We can concludein the same way as in dimension two that the critical shape Ω∗ is unstable. Notethat this argument is still valid if d = 2n+ 1, since the Hairy Sphere Theoremholds in such a case.

In conclusion, we claim that the only possible topologies for Ω∗ in dimen-sion 3 are domains with a smooth boundary of gender bigger or equal to onesuch as e.g., a torus. We restrict our attention to the case of critical shapesi.e., we assume that Ω∗ is a critical shape and evaluate the shape HessianD2LΛ∗(Ω∗; V ,V ).


3. The sign of shape Hessian at critical shapes

As it is the case in the classical theory of calculus of variations, the local extremesin constrained optimization problems imply the sign of the Hessian. We aregoing to verify the sign of the Hessian D2LΛ(Ω∗; V ,V ) on the kernel H of theconstraints. Since the volume constraints are imposed, H is the kernel of DVi.e.,

H =V ∈ V,

∫∂Ω∗

〈V ,n〉 = 0. (15)

Now, we evaluate the shape Hessian D2LΛ(Ω∗; V ,V ).First, we recall the useful lemma for derivation of the shape Hessian. Note

that the expression of the shape Hessian obtained via this lemma is not thesame as the canonical form given by the structure theorem at the critical point.The form derived below is convenient to deal with for the stability analysisperformed in Section 4.

Lemma 3.1 Let Ω be a domain in O, V a deformation field in V and g be asmooth function in C1(R2,R). Let J be a shape functional defined on O,

∀Ω ∈ O, J(Ω) =∫

∂Ω

g〈V ,n〉.

Then J is differentiable on O in the direction V and, moreover, the first ordershape derivative is given by

∀Ω ∈ O, DJ(Ω; V ) =∫

∂Ω

[g′ + div

(gV

)]〈V ,n〉.

Proof. It is just a game between the Hadamard derivation Lemma and Green’sformula. Actually, by Green’s formula, we have

J(Ω) =∫

∂Ω

g〈V ,n〉 =∫

Ω

div(gV

);

so an application of the Hadamard formula to the domain integral leads to

DJ(Ω; V ) =∫

Ω

div(g′V

)+ div

(div

(gV

)V

).

This expression can be rewritten in the form of a boundary integral

DJ(Ω; V ) =∫

∂Ω

[g′ + div

(gV

)]〈V ,n〉.


We recall the expression of the shape gradient for the volume constraints

DV(Ω; V ) =∫

∂Ω

〈V ,n〉.

By application of Lemma 3.1, we get

D2V(Ω; V ,V ) =∫

∂Ω

div(V

)〈V ,n〉. (16)

More delicate is application of Lemma 3.1 to the shape derivative of the energyfunctional DE(Ω,V ) since the appropriate element g is defined only on Ω. Sucha difficulty, which is common in the shape sensitivity analysis, does not implyany change of the related result of the lemma given in Dambrine (2000). Weget

D2E(Ω; V ,V ) =∫

∂Ω

〈∇u′,∇u〉〈V ,n〉+12

∫∂Ω

|∇u|2 div(V

)〈V ,n〉

+12

∫∂Ω

〈∇|∇u|2,V 〉〈V ,n〉.(17)

At the critical shape Ω∗, (11) holds and a straightforward computation leads to

D2LΛ(Ω∗; V ,V ) =∫

∂Ω∗〈∇u′,∇u〉〈V ,n〉+

12

∫∂Ω∗〈∇|∇u|2,V 〉〈V ,n〉. (18)

Since |∇u|2 is constant on ∂Ω∗ by (11), we have 〈∇|∇u|2,V 〉 = ∂n|∇u|2〈V ,n〉.As expected from the structure of the shape Hessian at a critical point, only thenormal component of V is present in the second order shape derivative of theLagrangian evaluated at Ω∗.

Now, the sign of Hessian is investigated on the hyperplane H. Let us splitthe Hessian computed in (17) into two parts, which are analysed separately,

A(t) =∫

∂Ωt

〈∇|∇u|2,V 〉〈V ,n〉, (19)

B(t) =∫

∂Ωt

〈∇u′,∇u〉〈V ,n〉. (20)

First, we consider A(0). The normal derivative ∂n|∇u|2 is related to thegeometry of ∂Ω∗ as it is expressed in the following lemma.

Lemma 3.2 Let K denote the Gauss curvature on ∂Ω∗. Then, we have

∂n|∇u|2 = 2 t(∇τu) K ∇τu,

furthermore, there exist two constants κmin and κmax such that

κmin |Λ∗| ‖〈V ,n〉‖L2(∂Ω∗) ≤12A(0) ≤ κmax |Λ∗| ‖〈V ,n〉‖L2(∂Ω∗).


Proof. The derivation of the above formula and of the estimates is based oncomputations in local coordinates. Let M be a given point on ∂Ω∗. There existsan orthogonal coordinate system in R3 such that locally around M = (0, 0, 0),∂Ω∗ admits the parametrisation z = f(x, y) where f is chosen in such a waythat f(0, 0) = fx(0, 0) = fy(0, 0) = 0. On ∂Ω∗, u is given by u

(x, y, f(x, y)

)and

the Neumann boundary condition leads to

〈∇u,n〉 =1√

1 + f2x + f2

y

ux

uy

uz

.

−fx

−fy

1

=

uxfx + uyfy − uz√1 + f2

x + f2y

= 0 on ∂Ω∗. (21)

Whence uxfx+uyfy−uz = 0 on ∂Ω∗, in particular, uz(M) = 0. We differentiate(21) with respect to x and y. Taking the values at M = (0, 0, 0) leads to

fxxux + fxyuy = uzx,

fxyux + fyyuy = uzy.(22)

We want to compute ∂n|∇u|2, that is

〈∇|∇u|2,n〉 =2√

1 + f2x + f2

y

uxuxx + uyuyx + uzuzx

uxuxy + uyuyy + uzuzy

uxuxz + uyuyz + uzuzz

.

−fx

−fy

1

.

For the point M , we get from (22)

〈∇|∇u|2,n〉 =2√

1 + f2x + f2

y

[ux

(fxxux + fxyuy

)+ uy

(fxyux + fyyuy

)],

=2√

1 + f2x + f2

y

(ux uy

) fxx fxy

fxy fyy

ux

uy

,

=2 t(∇τu) K ∇τu.

By compactness of ∂Ω∗, the extremal eigenvalues of the continuous Gauss cur-vature K on ∂Ω∗ attain the minimal and maximal values κmin and κmax, re-spectively. The constants κmin and κmax are determined by the minimal andthe maximal values of the algebraic curvatures of arcs traced on ∂Ω∗. Moreover,(11) relates the norm of ∇τuΩ∗ to the Lagrange multiplier Λ∗ so that

2|Λ∗|κmin ≤t (∇τuΩ∗) K ∇τuΩ∗ ≤ 2|Λ∗|κmax.


Remark 3.1 We cannot completely determine the quantity 〈∇|∇u|2,n〉 as weknow only two conditions for ∇u on ∂Ω∗ so it is not sufficient to determine thethree unknown coordinates. The situation is different for the Dirichlet boundarydatum, we refer e.g., to Descloux (1990) for details.

Now, we turn our attention to the term B(0). The main characteristic ofthis term is that it is negative, so it has a sign.

Lemma 3.3 There exists a constant C > 0 depending only of Ω∗ such that

∀V ∈ H, B(0) ≤ − C ‖〈V ,n〉‖2H1/2(∂Ω∗) . (23)

Proof. Since ∂nu = 0, then

B(0) =∫

∂Ω∗〈∇τu

′,∇τu〉〈V ,n〉

and an integration by parts leads to

B(0) = −∫

∂Ω∗u′ divτ (∇τu〉〈V ,n〉) = −

∫∂Ω∗

u′∂nu′.

Using the Green’s formula we obtain∫∂Ω∗

u′∂nu′ −

∫Ω∗u′∆u′ =

∫Ω∗|∇u′|2 ,

thus, B(0) = −‖u′‖2H1

0 (Ω∗). Since u′ solves (7), the elliptic regularity for weak

solution implies the first inequality below

‖u′‖H1(Ω∗) ≤ C1‖∂nu′‖H−1/2(∂Ω∗) ≤ C2‖〈V ,n〉∇τu‖H1/2(∂Ω∗)

≤ C3‖〈V ,n〉‖H1/2(∂Ω∗)

and we can justify the second and the third inequalities as follows. Since divτ (·)is a linear and continuous mapping from H1/2(∂Ω∗) into H−1/2(∂Ω∗), the sec-ond inequality holds. The third inequality can be deduced from the auxiliarymultiplier lemma proved in Dambrine (2000) for the space H1/2(∂Ω∗)×C1(∂Ω∗).The lemma can be used, since, from the elliptic regularity theory it follows that∇τu is C1 on ∂Ω∗. Obviously, the constants C1, C2 and C3 depend on thegeometrical attributes of the optimal shape Ω∗.

Conclusion. The sign of the Hessian D2LΛ(Ω∗; V ,V ) depends only of thegeometry of Ω∗. It is clear that the geometrical constraints to be satisfied(see Theorem 2.1) are relatively strong, whence we are unable at the moment topresent an explicit and simple example of the critical shape for which assumption(6) is verified. However, we have shown in this section that the principal term


of shape Hessian, that is B(0), is non-positive. Therefore, it seems reasonableto assume the existence of a critical point Ω∗ for the energy functional such thatthe necessary condition of non-negativity of D2LΛ(Ω∗; V ,V ) is fulfilled.

The second remark is fundamental from the point of view of stability analysisEven if the Hessian D2LΛ(Ω∗; V ,V ) is anticoercive, the related inequality holdsonly for the H1/2 norm and cannot be expected to be valid in the stronger normof the differentiability, which is actually the C2 norm. We are in the situationindicated by J. Descloux and have to find a remedy for this difficulty in order toperform the stability analysis. Therefore, we are going to prove the key estimate(5) which allows for positive results in the stability analysis.

4. Proof of the stability estimate (5)

In this section, we fix the critical shape Ω0 of E and a constant η > 0 whichdefines the maximal size in the norm C2,α of the perturbations which deformsthe shape Ω0. Let Θ denote an arbitrary diffeomorphism, an element of theball with the centre IR3 and of the radius η. We show in this section that thecondition D2LΛ(Ω0; V ,V ) ≥ c‖V ‖2

H1/2(∂Ω0)implies the strict inequality for the

energy functional E(Ω0) < E(Θ(Ω0)).The method proposed in Dambrine and Pierre (2000) and developed in

Dambrine (2002) consists of the geometrical part with the construction of apath Ω(t) connecting the critical shape Ω0 with the target shape Θ(Ω0), all ele-ments of the path belonging to the family of domains with the fixed volume v0.Such a construction is accomplished using the flow ΦΘ,t of the particular vectorfield XΘ, with the intermediate shapes Ω(t) given by the image ΦΘ,t(Ω0) of theshape Ω0. We consider the restriction of the shape functional to this path, thatis

eΘ(t) = E(Φt(Ω0)

). (24)

Then, the variations of the second derivative e′′Θ(t) of eΘ with respect to thevariable t can be estimated and the following result is established.

Proposition 4.1 There exists the modulus of continuity ω and the number η0

such that for all η ∈ (0, η0) and each volume preserving diffeomorphism Θ with‖Θ− IdRd‖3,α ≤ η,∣∣e′′Θ(t)− e′′Θ(0)

∣∣∣ ≤ Cω(η)‖〈XΘ,n〉‖2H1/2(∂Ω0) , (25)

for all t ∈ [0, 1].

For the proof of Proposition 4.1, the arguments used for the Dirichlet prob-lem in Dambrine and Pierre (2000) are adapted. We start with necessary prelim-inaries. The results of geometrical nature, which we use throughout the paper,are only reported. For the complete proofs we refer the reader to Dambrine(2000) in the bidimensional case and to Dambrine (2002) in the general case.


Preliminaries on the path construction within the family of admissi-ble shapes. To simplify the notation, the dependence on Θ is not indicatedwhenever Θ is fixed, which is the case in the second part of the section. How-ever, in the first part of the section we indicate the dependence on Θ, since weestablish the uniform estimates with respect to Θ. As it is shown in Dambrine(2002), the sufficient conditions for the existence of a local strict minimum forenergy functionals can be established by a construction of some specials pathswithin the admissible family of shapes. We use the same vector fields as thoseexploited in Dambrine (2002). We recall that, once an initial shape Ω0 and anadmissible perturbation Θ are chosen, then there exists the normal vector fieldXΘ such that

• the flow ΦΘ,t of the field XΘ at t = 1 maps Ω0 onto Θ(Ω0)• the field XΘ is divergence-free,• XΘ = mn, where n = ∇d∂Ω0/‖∇d∂Ω0‖ is an unitary extension of the nor-

mal field, and d∂Ω0 denotes the signed distance function to the boundary∂Ω0.

Moreover, the family ΦΘ,t enjoys the following properties proved in Dambrine(2002).

Proposition 4.2 (Variations of geometrical attributes) There exists a constantC > 0 such that for all t ∈ [0, 1]:

1. ‖DΦΘ,t − Id‖L∞ + ‖D2ΦΘ,t‖L∞ ≤ ‖ΦΘ,t − IdRd‖2,α;2. ‖DΦ−1

Θ,t − Id‖L∞ + ‖D[DΦ−1Θ,t]‖L∞ ≤ C‖ΦΘ,t − IdRd‖2,α;

3. Let J(t) be the Jacobian det(DΦΘ,t‖t(DΦΘ,t)−1n0‖

). Then

‖J(t)− 1‖L∞(∂Ω0) ≤ C‖ΦΘ,t − IdRd‖2,α,

‖J(t)− 1‖C1(∂Ω0) ≤ C‖ΦΘ,t − IdRd‖2,α;(26)

4. If nt denotes the unitary outer normal vector field on ΦΘ,t(∂Ω0),

‖nt ΦΘ,t − n0‖C1(∂Ω0) ≤ C‖ΦΘ,t − IdRd‖2,α . (27)

5. Moreover, if Θ has the C3-regularity and satisfies ‖Θ− IdRd‖3 ≤ 1/2,then we have the following estimate

‖nt ΦΘ,t − n0‖C2(∂Ω0) ≤ C‖ΦΘ,t − IdRd‖3 . (28)

Proposition 4.2 recalls the classical results valid for all smooth vector fieldsused for deformation of the initial shape. On the contrary, the following state-ment is specific for the chosen field XΘ = mn, since it can be verified by simplecalculations. Such a result is required, at least for technical reasons. It is notclear at the moment if conditions (29) can be relaxed for the problem underconsiderations.


Proposition 4.3 (Properties of XΘ) There is a constant C dependent only ofΩ0 such that

∀t ∈ [0, 1],

∥∥m ΦΘ,t −m

∥∥L2(∂Ω0)

≤ C‖m‖L2(∂Ω0)‖Θ− Id‖2,α,∥∥m ΦΘ,t −m∥∥

H1/2(∂Ω0)≤ C‖m‖H1/2(∂Ω0)‖Θ− Id‖2,α,∥∥m ΦΘ,t −m

∥∥H1(∂Ω0)

≤ C‖m‖H1(∂Ω0)‖Θ− Id‖2,α

(29)

where m is given by m = 〈XΘ, n〉.

In the previous works on the optimality conditions, the piecewise estimate upto the boundary of the second order derivatives of the state function along thepath were required. Such estimates can be provided by the classical Schaudertheory for the Dirichlet problem. Now, we mimic the same argument for theNeumann problem. We introduce the necessary notation and definitions. Theinverse transport operator is denoted by ΨΘ,t = (ΦΘ,t)−1, which allows forworking in the fixed domain setting, here the initial domain is Ω0. This createsan additional difficulty, i.e., the transported solution ut = ut ΦΘ,t (see belowfor the definition) does not solve any pure Neumann problem for the Laplacian.Instead, ut solves the boundary value problem for the perturbed operator L(t).We have the explicit form of the operator L(t),

L(t)v =[ n∑

i=1

n∑j=1

∂iΨαΘ,t∂jΨβ

Θ,t

]︸︷︷︸ ∂

2α,βv +

[ n∑i=1

∂2i,iΨ

αΘ,t

]︸︷︷︸ ∂αv ,

= aα,β(t) ∂2α,βv + bβ(t) ∂αv .

(30)

We use the simplified notation for Dt = tDΦ−1Θ,t, and for the transported

gradient

Dt∇ut = ∇(ut ΦΘ,t). (31)

If ut solves the problem −∆v = f in Ωt ,

〈∇v,n(t)〉 = 0 on ∂Ωt ,(32)

then the transported solution ut solves the following boundary value problem −L(t)v = f ΦΘ,t in Ω0 ,

〈Dt∇v,n(t) ΦΘ,t〉 = 0 on ∂Ω0 .(33)


The normal derivative corresponding to the Neumann boundary condition on∂Ωt is changed into a oblique derivative condition in the direction tDtn(t)ΦΘ,t

on ∂Ω0. Since n(t)Φt = Dtn/‖Dtn‖, the vector field can be expressed in termsof the outward normal field n on ∂Ω0,

tDtn(t) ΦΘ,t =tDtDtn

‖Dtn‖= ‖Dtn‖ B(t) n ,

see Section 6 for the definition of B(t).The first result we need is a uniform estimate in C2,α along the path, of the

transported state function.

Lemma 4.1 There exists a constant η0 > 0, which depends only of Ω0, and aconstant C = M(f), which depends only of η0, such that for each diffeomorphismΘ with the norm bounded ‖Θ − IdR2‖2,α ≤ η0 and for all t ∈ [0, 1], we have‖ut‖2,α,Ω0

≤M(f).

Proof. We adapt the proof given in Dambrine and Pierre (2000) to the case ofthe Neumann boundary conditions. First, we introduce the perturbed boundaryvalue problem (Pε,Θ,t) defined by −∆u+ εu = f in Ωt,

∂nu = 0 on ∂Ωt.(34)

Let uε denote the solution of the above equation, by (34) it follows that∫Ωt

uε =1ε

∫Ωt

f = 0.

It means that for uε the Poincare inequality can be used, and there exists aconstant CP , which depends only on the diameter of Ωt, and can be chosenuniformly with respect to Θ and t, such that

‖uε‖L2(Ωt) ≤ CP ‖∇uε‖L2(Ωt). (35)

We also introduce the transported solution utε = uε ΦΘ,t which solves −L(t)v + εv = f ΦΘ,t in Ω0,

〈Dt∇v,n(t) ΦΘ,t〉 = 0 on ∂Ω0.(36)

There exists the lower bound λ, uniform with respect to Θ and t, such that

λ|ξ|2 ≤ aα,β(t)ξαξβ .


From the definitions given in (4.2) and by Proposition 4.2(2), the coefficientsaα,β and bβ are bounded and there exists a constant M such that

‖aα,β(t)‖0,α,Ω0, ‖bβ(t)‖0,α,Ω0

≤M.

We also have to check that the co-normal direction n(t) is non tangential onthe boundary of Ω0 for t ∈ [0, 1]. This can be deduced from the properties ofthe normal vectors since

〈n(t),n〉 = 1− 〈n− n(t),n〉 ≥ θmin > 0.

This a direct application of Proposition 4.2. We use the Schauder estimate(see Gilbarg and Trudinger, 1983) to get the existence of a constant C =C(d, α, λ, θmin,Ω0) such that

‖utε‖2,α,Ω0

≤ C[‖ut

ε‖L∞(Ω0) + ‖f‖0,α

]. (37)

Note that by the assumption on the support of f , there is no need to precisethe domain where the Holder norm of f is considered. We use the classicalargument to find an upper bound for weak solutions of an elliptic equation (seeGilbarg and Trudinger, 1983) and get

‖utε‖L∞(Ω0) ≤ C

[‖f‖Lq(Ω0) + ‖ut

ε‖L2(Ω0)

].

One has now by the change of variable formula and the upper bound on DΦΘ,t

that

‖utε‖2

L2(Ω0) ≤∫

Ωt

(uε

)2∣∣D(ΦΘ,t)−1∣∣ ≤ C‖uε‖2

L2(Ωt)

After having multiplied (34) by uε, an integration by parts leads to

‖∇uε‖2L2(Ωt) + ε‖uε‖2

L2(Ωt) =∫

Ωt

fuε

and we obtain

‖∇uε‖2L2(Ωt) ≤ ‖f‖L2(Ωt) ‖uε‖L2(Ωt).

From the Poincare inequality, one gets

‖uε‖2L2(Ωt) ≤ CP ‖f‖L2(Ωt) ‖uε‖L2(Ωt) ⇒ ‖uε‖L2(Ωt) ≤ CP ‖f‖L2(Ωt)

hence we can deduce the estimate independent of ε

‖utε‖2,α,Ω0

≤M(f),


where M(f) is a constant that depends only on M and η. We pass to the limitwith ε→ 0 and obtain the required estimate

‖ut‖2,α,Ω0≤M(f) .

Then, we can conclude, in the same way as in Dambrine and Pierre (2000),by the compactness of the embedding of the space C2,α(Ω0) into C2(Ω0) that

ω(η) := supt∈[0,1],‖Θ−IdRd‖2,α≤η

‖ut − u0‖C2(Ω0) → 0 with η 0.

This means that the function ω is a modulus of continuity. The resulting con-tinuity in C2(Ω0) of the transported state function is given in the followingproposition.

Proposition 4.4 There exists a modulus of continuity ω such that for all Θwith ‖Θ− IdRd‖2,α ≤ η and all t in [0, 1] one has

‖ut − u0‖C2(Ω0) ≤ ω(η). (38)

Using the previous result on the shape differentiability of the functionalE, we can deduce easily that eΘ is twice differentiable, with the second orderderivative of the form

e′′Θ(t) =12

∫∂Ωt

〈∇|∇u|2,V 〉〈V ,n〉+∫

∂Ωt

〈∇u′,∇u〉〈V ,

=12A(t) +B(t).

We have the simplified expression, since by construction div(XΘ

)= 0.

Analysis of A(t)−A(0). First, we recall the definition of A,

A(t) =∫

∂Ωt

m2〈∇|∇u|2, n〉〈n,n(t)〉.

Whence

A(t)−A(0) =∫

∂Ω0

m2〈Dt∇|Dt∇ut|2,n〉〈n,n(t)Φt〉J(t)−m2〈∇|∇u0|2,n〉.

Let us denotec1(t) := 〈Dt∇|Dt∇ut|2,n〉,

c2(t) := 〈n,n(t) Φt〉,

c3(t) := J(t).


We can conclude as in Dambrine and Pierre (2000), for i = 1, 2, 3 we verify that

|ci(t)| ≤ C and |ci(t)− ci(0)| ≤ Cω(η) .

Then, Proposition 4.3 and simple calculations show,

|A(t)−A(0)| =∣∣ ∫

∂Ω0

m2Π3i=1ci(t)−m2Π3

i=1ci(0)∣∣ ≤ Cω(η)‖m‖2

L2(∂Ω0). (39)

Analysis of B(t)−B(0). First, we rewrite the term B(t) in the same way asin the proof of Lemma 3.3, hence

B(t) =∫

Ωt

|∇u′(t)|2 ,

where u′(t) solves −∆v = 0 in Ωt,

∂nv = divτ (〈XΘ,n〉∇τu) := N(0) on ∂Ωt.

After the transport of the domain integral to the critical domain Ω0, we get thedifference (the Jacobian DΦt ≡ 1 since XΘ is divergence-free)

B(t)−B(0) =∫

Ω0

|Dt∇(ut)′|2 − |∇u′(0)|2 (40)

where (ut)′ := u′(t) Φt and solves −L(t)v = 0 in Ω0,

〈∇v, ‖Dtn‖ B(t) n〉 = N(t) on ∂Ωt ,

with N(t) defined below, we refer to Section 6 for a justification of the transportformulae (49), (50) and for the definitions of the functions α(x, t) and β(x, t)),

N(t) := α(x, t)m+ 〈β(x, t),∇m〉. (41)

What is important here is not the exact form of the functions α and β, thecomputation of the functions is postponed to the end of the section, but thestructure and the regularity of both functions. The classical arguments of ellipticregularity can be used to show that α is C1,α while β is C2,α. Now, we use theclassical elliptic estimates :

‖(ut)′‖H1(Ω0) ≤ C‖N(t)‖H−1/2(∂Ω0)

≤ C[‖α(x, t)m‖H−1/2(∂Ω0) + ‖〈β(x, t),∇m〉‖H−1/2(∂Ω0).

We now use the following product estimate deduced, by duality, from the H1/2×C1 multiplier lemma of Dambrine and Pierre (2000).


Lemma 4.2 (Multiplier-product estimate H−1/2 × C1 )If ψ ∈ H−1/2(∂Ω0) ∩ C0(∂Ω0) and f ∈ C1, then there exist a constant C(Ω0)such that

‖fψ‖H−1/2(∂Ω0) ≤ C(Ω0)‖f‖C1(∂Ω0)‖ψ‖H−1/2(∂Ω0). (42)

Proof. If ψ is C0, then by definition

‖fψ‖H−1/2(∂Ω0) = sup‖ϕ‖

H1/2(∂Ω0)≤1

∫∂Ω0

fψϕ.

Therefore, for all ϕ ∈ H1/2(∂Ω0) with ‖ϕ‖H1/2(∂Ω0) = 1, the duality scalarproduct is well defined and bounded,

〈fψ, ϕ〉 =∫

∂Ω0

fψϕ =∫

∂Ω0

ψfϕ ,

≤ ‖ψ‖H−1/2(∂Ω0)‖fϕ‖H1/2(∂Ω0) ,

≤ ‖ψ‖H−1/2(∂Ω0)C(Ω0)‖f‖C1(∂Ω0)‖ϕ‖H1/2(∂Ω0)

which implies (42).

Remark 4.1 Note that the corresponding C0 lemma is not true (at least, cannotbe shown as easily), because the dual estimation does not have any sense. Theproduct of a continuous function with an element of H1/2(∂Ω0) has no reasonto belong to the trace space H1/2. That is why we consider only C3,α shapes.

Then, we use Lemma 4.2 to show that

‖(ut)′‖H1(Ω0) ≤ C[‖α(x, t)‖C1‖m‖H−1/2(∂Ω0) + ‖β(x, t)‖C1‖∇m‖H−1/2(∂Ω0) ,

≤ C[‖α(x, t)‖C1‖m‖H−1/2(∂Ω0) + ‖β(x, t)‖C1‖m‖H1/2(∂Ω0) ,

≤ C‖m‖H1/2(∂Ω0) .

We are going to verify∣∣(ut)′ − u′0∣∣ ≤ Cω(η)‖m‖H1/2(∂Ω0) . (43)

The difference v = (ut)′ − u′0 solves the equation

−L(t)((ut)′ − u′0) = L(t)u′0 = (L(t)−∆)u′0 .

The boundary condition is a little bit more difficult to get since the direction ofderivation is different, hence

〈∇((ut)′ − u′0), ‖Dtn‖ B(t) n〉 = N(t)− 〈∇u′0, ‖Dtn‖ B(t) n〉︸︷︷︸,〈∇u′0, n〉+ 〈∇u′0, (‖Dtn‖ B(t)− Id)n〉

= [N(t)−N(0)]− 〈∇u′0, [‖Dtn‖ B(t)− Id]n〉.


From Proposition 4.2 it follows that

‖〈∇u′0, [‖Dtn‖ B(t)− Id]n〉‖H−1/2(∂Ω0) ≤ Cω(η)‖m‖H1/2(∂Ω0) .

On the other hand

N(t)−N(0) = α(x, t)m− α(x, 0)m+ 〈β(x, t),∇m〉 − 〈β(x, 0),∇m〉,= [α(x, t)− α(x, 0)]m+ α(x, 0)[m−m]

+ 〈[β(x, t)− β(x, 0)],∇m〉+ 〈β(x, 0),∇[m−m]〉.

From the latter expression, we can deduce by (45) and using Proposition 4.3that

‖N(t)−N(0)‖H−1/2(∂Ω0) ≤ Cω(η)‖m‖H1/2(∂Ω0) .

As in Dambrine and Pierre (2000) and Dambrine (2000), we get easily that

‖[L(t)−∆]u′0‖H−1(Ω0) ≤ Cω(η)‖m‖H1/2(∂Ω0) .

We conclude from the classical estimates in Sobolev spaces that

‖(ut)′ − u′0‖H1(Ω0) ≤ C[‖[L(t)−∆]u′0‖H−1(Ω0)+

‖〈∇((ut)′ − u′0), ‖Dtn‖ B(t) n〉‖H−1/2(∂Ω0)

],

≤ C ω(η)‖m‖H1/2(∂Ω0)

Therefore, we deduce from (40) that∣∣B(t)−B(0)∣∣∣ ≤ Cω(η)‖m‖2

H1/2(∂Ω0). (44)

Computation of α and β: Finally, we determine the expressions for α(x, t)and β(x, t). From (49), the transported tangential gradient is given by(

∇τ ,tu(t)) Φt = Dt∇ut − 〈∇ut,B(t)n〉Dtn .

Hence, we get with D2tt = DDt and µ :=

[〈XΘ,n(t)〉

] Φt = m〈n,n(t) ΦΘ,t〉

that ∇(µφ) = DΦt(∇µ)Φt. We make use of the formula Dav = aDv+v t∇ato obtain

D[(〈XΘ,n(t)〉∇τ ,tu(t)

) Φt

]=

[Dt∇ut − 〈∇ut,B(t)n〉Dtn

]t∇µ+ µD


]=


]t∇µ+ µ

[−Dtn

t∇(〈∇ut,B(t)n〉

)+ D2

tt∇ut + (Dt)2D2ut − 〈∇ut,B(t)n〉(D2ttn + DtDn)

].


We then use (50), and the equality Tr(vt1v2

)= 〈v1,v2〉, to get

N(t) = 〈∇µ, (Dt)2∇ut〉 − 〈∇ut,B(t)n〉〈∇µ, (Dt)2n〉+ µTr

(D2

tt∇ut + (Dt)2D2ut)

− µ〈∇ut,B(t)n〉[Tr

(D2

ttn + DtDn)]− µ〈∇〈∇ut,B(t)n〉,Dtn〉

−[〈∇µ,n〉

[〈Dt∇ut,B(t)n〉 − 〈∇ut,B(t)n〉〈Dtn,B(t)n〉

]− µ

(〈Dtn,B(t)n〉〈∇〈∇ut,B(t)n〉,n〉+ 〈(D2

tt∇ut)n,B(t)n〉

+ 〈(Dt)2D2utn,B(t)n〉

− 〈∇ut,B(t)n〉〈(D2ttn + DtDn)n,B(t)n〉

)].

By the chain rule, ∇µ = mDΦt∇〈n,n(t) Φt〉+ 〈n,n(t)〉DΦt∇m, as a result

α(x, t) = 〈DΦt∇〈n,n(t) Φt〉,[(Dt)2∇ut + 〈∇ut,B(t)n〉(Dt)2n

+[〈∇ut,B(t)n〉〈Dtn,B(t)n〉 − 〈Dt∇ut,B(t)n〉

]n

]〉

+〈n,n(t)〉[Tr

(D2

tt∇ut + (Dt)2D2ut)

+ 〈∇ut,B(t)n〉

Tr(D2

ttn + DtDn)− 〈∇〈∇ut,B(t)n〉,Dtn〉

+(〈Dtn,B(t)n〉〈∇〈∇ut,B(t)n〉,n〉+ 〈(D2

tt∇ut)n,B(t)n〉

+〈(Dt)2D2utn,B(t)n〉

−〈∇ut,B(t)n〉〈(D2ttn + DtDn)n,B(t)n〉

)]β(x, t) = 〈n,n(t) Φt〉Dt

[(Dt)2∇ut − 〈∇ut,B(t)n〉(Dt)2n

+[〈∇ut,B(t)n〉〈Dtn,B(t)n〉 − 〈Dt∇ut,B(t)n〉

]n

].

The important observation is that α(x, t) and β(x, t) are C1 functions and thereexists C > 0 such that for all t ∈ [0, 1]

‖α(x, t)‖C∞ , ‖β(x, t)‖C1 ≤ C , (45)‖α(x, t)− α(x, 0)‖C∞ , ‖β(x, t)− β(x, 0)‖C1 ≤ Cω(η) .

To prove this, we observe that the inequalities in (45) are stable under themultiplication and the addition (see Dambrine, 2000). This means that if α1

and α2 satisfy (45) then it is also the case for the product α1.α2 and the sumα1 + α2. Hence, (45) follows from Propositions 4.2 and 4.4.

5. Analysis of the energy functional Eσ

Shape derivatives of the perimeter. We recall without proofs some clas-sical results on the functional P(Ω) = Hd−1(∂Ω). This functional P is twice


differentiable with the shape derivatives given by

DP(Ω; V ) =∫

∂Ω

H〈V ,n〉,

D2P(Ω; V ,V ′) =∫∂Ω

〈∇τ 〈V ,n〉,∇τ 〈V ′,n〉+ 〈V ,n〉〈V ′,n〉[H2 − Tr

(tDnDn

)]+

∫∂Ω

H[V τ .Dn.V ′

τ + n.DV .V ′τ + n.DV ′.V τ

].

(46)

Here, H denotes the mean curvature of ∂Ω. An important property proved inDambrine (2000) is the estimate for the second order variation D2P(Ω; V ,V ),similar to (5). The estimate takes the form

|P ′′Θ(t)− P ′′

Θ(0)| ≤ ω(‖Θ− Id‖2,α)‖〈V ,n〉‖H1(∂Ω). (47)

Euler equation for Eσ. From the results of Section 2 and (46), we obtainthat any critical shape Ω∗

σ, which minimizes the functional Eσ over the class ofdomains with fixed volume, v0 satisfies the following Euler-Lagrange equation

12|∇τuΩ∗σ |+ σH + Λ∗ = 0. (48)

We remark that the perimeter term in Eσ suppresses the argument used for thecancellation of the Lagrange multiplier Λ∗. Therefore, there is no obstructionfor the existence of the stable critical shapes.

6. Transport of differential operators

This section is devoted to the justification of the formulae we have used in thepaper for the transport of tangential differential operators. The starting pointis the formula (31) valid for the gradient. Following Soko lowski and Zolesio(1992), one can deduce the formula for the tangential gradient. We introducethe notation. Let ρ denote a C2 function defined on Rd and let A be a C2

vector field. Assume that ∂Ω is a C2 manifold. Suppose that V is a C2 vectorfield, and let Φt be the flow of V , with ∂Ωt = Φt(∂Ω). We denote by ρt (resp.At) the restriction of ρ (resp. A) to ∂Ωt. Let ∇τ ,t (resp. divτ ,t (·)) denotethe tangential gradient (resp. divergence) on ∂Ωt. The tangential gradient isdefined as

∇τ ,tρt := ∇ρ− 〈∇ρ,n(t)〉n(t).

Moreover, we know that(∇ρ) Φt = Dt∇(ρ Φt),

n(t) Φt =Dtn

‖Dtn‖.


Therefore, we have

(∇τ ,tρt) Φt = Dt∇(ρ Φt)− 〈Dt∇(ρ Φt),Dtn

‖Dtn‖〉

Dtn

‖Dtn‖,

= Dt

[∇(ρ Φt)− 〈∇(ρ Φt),

tDtDtn

‖Dtn‖2〉n

].

We follow the notations of Soko lowski and Zolesio (1992) and set

B(t) =tDtDt

‖Dtn‖2.

B(t) is a symmetric matrix function such that B(0) is the identity and 〈B(t)n,n〉= 1 for all t and all Θ. Note also that the leading part of the transported differ-ential operator L(t) is given by the matrix A(t) =

(aα,β(t)

)defined in (30), and

is related to B(t) by A(t) = ‖Dtn‖2B(t). Therefore, we obtain the followingformula for the transported tangential gradient:

(∇τ ,tρt) Φt = Dt∇(ρ Φt)− 〈∇(ρ Φt),B(t)n〉Dtn . (49)

The tangential divergence is defined by

divτ ,t (At) := div(A

)− 〈DAn(t),n(t)〉.

Hence, we get

(divτ ,t (At)) Φt = Tr(DΦ−1

t D (A Φt))− 〈D (A Φt)

Dtn

‖Dtn‖,

Dtn

‖Dtn‖〉.

We have obtained the formula for the transport of the tangential divergence:

(divτ ,t (At)) Φt = Tr(

tDtD (A Φt))− 〈D (A Φt) n,B(t)n〉. (50)

References

Belov, S. and Fuji, N. (1997) Symmetry and sufficient condition of optimal-ity in a domain optimization problem. Control and Cybernetics, 26(1)45–56.

Courant, R.and Hilbert, D. (1989)Methods of mathematical physics. Vol.II. Partial differential equations. Reprint of the 1962 original. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

Dambrine, M.(2002)About the variations of the shape Hessian and sufficientconditions of stability for critical shapes. Revista Real Academia Ciencias,RACSAM, 96(1).

Dambrine, M. (2000) Hessiennes de forme et stabilite des formes critiques.PhD thesis, Ecole Normale Superieure de Cachan,


Dambrine, M. and Pierre, M. (2000)About stability of equilibrium shapes.Mathematical Modelling and Numerical Analysis, 34(4), 811–834.

Descloux, J. (1990) The bidimensionnal shaping problem without surfacetension. Technical report, Ecole Polytechnique Federale de Lausanne.

Eppler., K. (2000) Second derivatives and suffient optimality conditions forshape functionals. Control and Cybernetics, 29(2), 485–511.

Gilbarg, D. and Trudinger, N.S. (1983) Elliptic partial differential equa-tions of second order, Springer-Verlag, Berlin, second edition.

Henrot, A. and Pierre, M. (1989) Un probleme inverse en formage desmetaux liquides. RAIRO Model. Math. Anal. Numer., 23(1), 155–177.

Malanowski, K. (2001)Stability and sensitivity analysis for optimal controlproblems with control-state constraints. Dissertationes Math. (RozprawyMat.) 394.

Murat, F. and Simon J. (1977) Optimal control with respect to the do-main. These de l’Universite Paris VI.

Soko lowski, J. and Zolesio, J.-P. (1992) Shape sensitivity analysis. In-troduction to shape optimization. Springer-Verlag, Berlin

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